Производная x^sin(x^x)

Вы ввели

    / x\
 sin\x /
x       

$$x^{\sin{\left (x^{x} \right )}}$$

Подробное решение

1. Не могу найти шаги в поиске этой производной.

   Но производная

   $\left(\log{\left (\sin{\left (x^{x} \right )} \right )} + 1\right) \sin^{\sin{\left (x^{x} \right )}}{\left (x^{x} \right )}$

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Ответ:

$\left(\log{\left (\sin{\left (x^{x} \right )} \right )} + 1\right) \sin^{\sin{\left (x^{x} \right )}}{\left (x^{x} \right )}$

Первая производная

    / x\ /   / x\                                 \
 sin\x / |sin\x /    x                 / x\       |
x       *|------- + x *(1 + log(x))*cos\x /*log(x)|
         \   x                                    /

$$x^{\sin{\left (x^{x} \right )}} \left(x^{x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} \cos{\left (x^{x} \right )} + \frac{1}{x} \sin{\left (x^{x} \right )}\right)$$

Вторая производная

         /                                          2                                                                                                                                \
    / x\ |/   / x\                                 \       / x\    x    / x\                                                                                   x                 / x\|
 sin\x / ||sin\x /    x                 / x\       |    sin\x /   x *cos\x /*log(x)    x             2    / x\           2*x             2           / x\   2*x *(1 + log(x))*cos\x /|
x       *||------- + x *(1 + log(x))*cos\x /*log(x)|  - ------- + ----------------- + x *(1 + log(x)) *cos\x /*log(x) - x   *(1 + log(x)) *log(x)*sin\x / + -------------------------|
         |\   x                                    /        2             x                                                                                             x            |
         \                                                 x                                                                                                                         /

$$x^{\sin{\left (x^{x} \right )}} \left(- x^{2 x} \left(\log{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} \sin{\left (x^{x} \right )} + x^{x} \left(\log{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} \cos{\left (x^{x} \right )} + \left(x^{x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} \cos{\left (x^{x} \right )} + \frac{1}{x} \sin{\left (x^{x} \right )}\right)^{2} + \frac{2 x^{x}}{x} \left(\log{\left (x \right )} + 1\right) \cos{\left (x^{x} \right )} + \frac{x^{x}}{x} \log{\left (x \right )} \cos{\left (x^{x} \right )} - \frac{1}{x^{2}} \sin{\left (x^{x} \right )}\right)$$

Третья производная

         /                                          3                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            \
    / x\ |/   / x\                                 \         / x\     /   / x\                                 \ /     / x\    x    / x\                                                                                   x                 / x\\      x    / x\                                      x    / x\                                                 2*x             2    / x\      x                 / x\                                            x             2    / x\      2*x                        / x\      x                 / x\       |
 sin\x / ||sin\x /    x                 / x\       |    2*sin\x /     |sin\x /    x                 / x\       | |  sin\x /   x *cos\x /*log(x)    x             2    / x\           2*x             2           / x\   2*x *(1 + log(x))*cos\x /|   3*x *cos\x /    x             3    / x\          x *cos\x /*log(x)    3*x             3    / x\          3*x   *(1 + log(x)) *sin\x /   3*x *(1 + log(x))*cos\x /      2*x             3           / x\   3*x *(1 + log(x)) *cos\x /   3*x   *(1 + log(x))*log(x)*sin\x /   3*x *(1 + log(x))*cos\x /*log(x)|
x       *||------- + x *(1 + log(x))*cos\x /*log(x)|  + --------- + 3*|------- + x *(1 + log(x))*cos\x /*log(x)|*|- ------- + ----------------- + x *(1 + log(x)) *cos\x /*log(x) - x   *(1 + log(x)) *log(x)*sin\x / + -------------------------| + ------------ + x *(1 + log(x)) *cos\x /*log(x) - ----------------- - x   *(1 + log(x)) *cos\x /*log(x) - ---------------------------- - ------------------------- - 3*x   *(1 + log(x)) *log(x)*sin\x / + -------------------------- - ---------------------------------- + --------------------------------|
         |\   x                                    /         3        \   x                                    / |      2             x                                                                                             x            |         2                                                   2                                                           x                              2                                                                x                                x                                   x                |
         \                                                  x                                                    \     x                                                                                                                         /        x                                                   x                                                                                          x                                                                                                                                                       /

$$x^{\sin{\left (x^{x} \right )}} \left(- x^{3 x} \left(\log{\left (x \right )} + 1\right)^{3} \log{\left (x \right )} \cos{\left (x^{x} \right )} - 3 x^{2 x} \left(\log{\left (x \right )} + 1\right)^{3} \log{\left (x \right )} \sin{\left (x^{x} \right )} + x^{x} \left(\log{\left (x \right )} + 1\right)^{3} \log{\left (x \right )} \cos{\left (x^{x} \right )} + \left(x^{x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} \cos{\left (x^{x} \right )} + \frac{1}{x} \sin{\left (x^{x} \right )}\right)^{3} + 3 \left(x^{x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} \cos{\left (x^{x} \right )} + \frac{1}{x} \sin{\left (x^{x} \right )}\right) \left(- x^{2 x} \left(\log{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} \sin{\left (x^{x} \right )} + x^{x} \left(\log{\left (x \right )} + 1\right)^{2} \log{\left (x \right )} \cos{\left (x^{x} \right )} + \frac{2 x^{x}}{x} \left(\log{\left (x \right )} + 1\right) \cos{\left (x^{x} \right )} + \frac{x^{x}}{x} \log{\left (x \right )} \cos{\left (x^{x} \right )} - \frac{1}{x^{2}} \sin{\left (x^{x} \right )}\right) - \frac{3}{x} x^{2 x} \left(\log{\left (x \right )} + 1\right)^{2} \sin{\left (x^{x} \right )} - \frac{3}{x} x^{2 x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} \sin{\left (x^{x} \right )} + \frac{3 x^{x}}{x} \left(\log{\left (x \right )} + 1\right)^{2} \cos{\left (x^{x} \right )} + \frac{3 x^{x}}{x} \left(\log{\left (x \right )} + 1\right) \log{\left (x \right )} \cos{\left (x^{x} \right )} - \frac{3 x^{x}}{x^{2}} \left(\log{\left (x \right )} + 1\right) \cos{\left (x^{x} \right )} - \frac{x^{x}}{x^{2}} \log{\left (x \right )} \cos{\left (x^{x} \right )} + \frac{3 x^{x}}{x^{2}} \cos{\left (x^{x} \right )} + \frac{2}{x^{3}} \sin{\left (x^{x} \right )}\right)$$